Properties

Label 630f
Number of curves 8
Conductor 630
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("630.f1")
sage: E.isogeny_class()

Elliptic curves in class 630f

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
630.f7 630f1 [1, -1, 0, -369, 1053] 2 384 \(\Gamma_0(N)\)-optimal
630.f5 630f2 [1, -1, 0, -3249, -69795] 4 768  
630.f4 630f3 [1, -1, 0, -24129, 1448685] 6 1152  
630.f2 630f4 [1, -1, 0, -51849, -4531275] 2 1536  
630.f6 630f5 [1, -1, 0, -729, -177147] 2 1536  
630.f3 630f6 [1, -1, 0, -24309, 1426113] 12 2304  
630.f1 630f7 [1, -1, 0, -58059, -3373137] 6 4608  
630.f8 630f8 [1, -1, 0, 6561, 4778595] 6 4608  

Rank

sage: E.rank()

The elliptic curves in class 630f have rank \(0\).

Modular form 630.2.a.f

sage: E.q_eigenform(10)
\( q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} + 2q^{13} - q^{14} + q^{16} + 6q^{17} - 4q^{19} + O(q^{20}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels.